# If $k$ is any natural number, then does there exist a sequence $(p_n)$ of primes such that for each $n \in \mathbb{N}$, $k|(p_n +1)$?

If $k$ is any natural number, then does there exist a sequence $(p_n)$ of primes such that for each $n \in \mathbb{N}$, $k|(p_n +1)$?

It would be equally good if one can prove/disprove the above question with the condition that $k|(p_n-1)$ instead of $k|(p_n+1)$.

Your condition I is just to say that $p_n\equiv -1\pmod k$, and condition II: $p_n\equiv 1\pmod k$. By Dirichlet theorem, this is true in any case.